Logic: Deductive and Inductive Part 4

_Truth will prevail_ means that _All true propositions are accepted at last_ (by sheer force of being true, in spite of interests, prejudices, ignorance and indifference). To bear this in mind may make one cautious in the use of abstract terms.

In the above paragraphs a distinction is implied between Singular and Distributive Universals; but, technically, every term, whether subject or predicate, when taken in its full denotation (or universally), is said to be 'distributed,' although this word, in its ordinary sense, would be directly applicable only to general terms. In the above examples, then, 'Queen,' 'Black Watch,' 'apes,' and 'truth' are all distributed terms. Indeed, a simple definition of the Universal Proposition is 'one whose subject is distributed.'

A Particular Proposition is one that has a general term for its subject, whilst its predicate is not affirmed or denied of everything the subject denotes; in other words, it is one whose subject is not distributed: as _Some lions inhabit Africa_.

In ordinary discourse it is not always explicitly stated whether predication is universal or particular; it would be very natural to say _Lions inhabit Africa_, leaving it, as far as the words go, uncertain whether we mean _all_ or _some_ lions. Propositions whose quant.i.ty is thus left indefinite are technically called 'preindesignate,' their quant.i.ty not being stated or designated by any introductory expression; whilst propositions whose quant.i.ty is expressed, as _All foundling-hospitals have a high death-rate_, or _Some wine is made from grapes_, are said to be 'predesignate.' Now, the rule is that preindesignate propositions are, for logical purposes, to be treated as particular; since it is an obvious precaution of the science of proof, in any practical application, _not to go beyond the evidence_. Still, the rule may be relaxed if the universal quant.i.ty of a preindesignate proposition is well known or admitted, as in _Planets shine with reflected light_--understood of the planets of our solar system at the present time. Again, such a proposition as _Man is the paragon of animals_ is not a preindesignate, but an abstract proposition; the subject being elliptical for _Man according to his proper nature_; and the translation of it into a predesignate proposition is not _All men are paragons_; nor can _Some men_ be sufficient, since an abstract can only be adequately rendered by a distributed term; but we must say, _All men who approach the ideal_. Universal real propositions, true without qualification, are very scarce; and we often subst.i.tute for them _general_ propositions, saying perhaps--_generally, though not universally, S is P_. Such general propositions are, in strictness, particular; and the logical rules concerning universals cannot be applied to them without careful scrutiny of the facts.

The marks or predesignations of Quant.i.ty commonly used in Logic are: for Universals, _All_, _Any_, _Every_, _Whatever_ (in the negative _No_ or _No one_, see next --); for Particulars, _Some_.

Now _Some_, technically used, does not mean _Some only,_ but _Some at least_ (it may be one, or more, or all). If it meant '_Some only_,'

every particular proposition would be an exclusive exponible (chap. ii.

-- 3); since _Only some men are wise_ implies that _Some men are not wise_. Besides, it may often happen in an investigation that all the instances we have observed come under a certain rule, though we do not yet feel justified in regarding the rule as universal; and this situation is exactly met by the expression _Some_ (_it may be all_).

The words _Many_, _Most_, _Few_ are generally interpreted to mean _Some_; but as _Most_ signifies that exceptions are known, and _Few_ that the exceptions are the more numerous, propositions thus predesignate are in fact exponibles, mounting to _Some are_ and _Some are not_. If to work with both forms be too c.u.mbrous, so that we must choose one, apparently _Few are_ should be treated as _Some are not_.

The scientific course to adopt with propositions predesignate by _Most_ or _Few_, is to collect statistics and determine the percentage; thus, _Few men are wise_--say 2 per cent.

The Quant.i.ty of a proposition, then, is usually determined entirely by the quant.i.ty of the subject, whether _all_ or _some_. Still, the quant.i.ty of the predicate is often an important consideration; and though in ordinary usage the predicate is seldom predesignate, Logicians agree that in every Negative Proposition (see -- 2) the predicate is 'distributed,' that is to say, is denied altogether of the subject, and that this is involved in the form of denial. To say _Some men are not brave_, is to declare that the quality for which men may be called brave is not found in any of the _Some men_ referred to: and to say _No men are proof against flattery_, cuts off the being 'proof against flattery'

entirely from the list of human attributes. On the other hand, every Affirmative Proposition is regarded as having an undistributed predicate; that is to say, its predicate is not affirmed exclusively of the subject. _Some men are wise_ does not mean that 'wise' cannot be predicated of any other beings; it is equivalent to _Some men are wise_ (_whoever else may be_). And _All elephants are sagacious_ does not limit sagacity to elephants: regarding 'sagacious' as possibly denoting many animals of many species that exhibit the quality, this proposition is equivalent to '_All elephants are_ some _sagacious animals_.' The affirmative predication of a quality does not imply exclusive possession of it as denial implies its complete absence; and, therefore, to regard the predicate of an affirmative proposition as distributed would be to go beyond the evidence and to take for granted what had never been alleged.

Some Logicians, seeing that the quant.i.ty of predicates, though not distinctly expressed, is recognised, and holding that it is the part of Logic "to make explicit in language whatever is implicit in thought,"

have proposed to exhibit the quant.i.ty of predicates by predesignation, thus: 'Some men are _some_ wise (beings)'; 'some men are not _any_ brave (beings)'; etc. This is called the Quantification of the Predicate, and leads to some modifications of Deductive Logic which will be referred to hereafter. (See -- 3; chap. vii. -- 4, and chap. viii. -- 3.)

-- 2. As to Quality, Propositions are either Affirmative or Negative. An Affirmative Proposition is, formally, one whose copula is affirmative (or, has no negative sign), as _S--is--P, All men--are--partial to themselves_. A Negative Proposition is one whose copula is negative (or, has a negative sign), as _S--is not--P, Some men--are not--proof against flattery_. When, indeed, a Negative Proposition is of Universal Quant.i.ty, it is stated thus: _No S is P, No men are proof against flattery_; but, in this case, the detachment of the negative sign from the copula and its a.s.sociation with the subject is merely an accident of our idiom; the proposition is the same as _All men--are not--proof against flattery_. It must be distinguished, therefore, from such an expression as _Not every man is proof against flattery_; for here the negative sign really restricts the subject; so that the meaning is--_Some men at most_ (it may be _none) are proof against flattery_; and thus the proposition is Particular, and is rendered--_Some men--are not--proof against flattery_.

When the negative sign is a.s.sociated with the predicate, so as to make this an Infinite Term (chap. iv. -- 8), the proposition is called an Infinite Proposition, as _S is not-P_ (or _p), All men are--incapable of resisting flattery_, or _are--not-proof against flattery_.

Infinite propositions, when the copula is affirmative, are formally, themselves affirmative, although their force is chiefly negative; for, as the last example shows, the difference between an infinite and a negative proposition may depend upon a hyphen. It has been proposed, indeed, with a view to superficial simplification, to turn all Negatives into Infinites, and thus render all propositions Affirmative in Quality. But although every proposition both affirms and denies something according to the aspect in which you regard it (as _Snow is white_ denies that it is any other colour, and _Snow is not blue_ affirms that it is some other colour), yet there is a great difference between the definite affirmation of a genuine affirmative and the vague affirmation of a negative or infinite; so that materially an affirmative infinite is the same as a negative.

Generally Mill's remark is true, that affirmation and denial stand for distinctions of fact that cannot be got rid of by manipulation of words.

Whether granite sinks in water, or not; whether the rook lives a hundred years, or not; whether a man has a hundred dollars in his pocket, or not; whether human bones have ever been found in Pliocene strata, or not; such alternatives require distinct forms of expression. At the same time, it may be granted that many facts admit of being stated with nearly equal propriety in either Quality, as _No man is proof against flattery_, or _All men are open to flattery_.

But whatever advantage there is in occasionally changing the Quality of a proposition may be gained by the process of Obversion (chap. vii. -- 5); whilst to use only one Quality would impair the elasticity of logical expression. It is a postulate of Logic that the negative sign may be transferred from the copula to the predicate, or from the predicate to the copula, without altering the sense of a proposition; and this is justified by the experience that not to have an attribute and to be without it are the same thing.

-- 3. A. I. E. O.--Combining the two kinds of Quant.i.ty, Universal and Particular, with the two kinds of Quality, Affirmative and Negative, we get four simple types of proposition, which it is usual to symbolise by the letters A. I. E. O., thus:

A. Universal Affirmative -- All S is P.

I. Particular Affirmative -- Some S is P.

E. Universal Negative -- No S is P.

O. Particular Negative -- Some S is not P.

As an aid to the remembering of these symbols we may observe that A. and I. are the first two vowels in _affirmo_ and that E. and O. are the vowels in _nego_.

It must be acknowledged that these four kinds of proposition recognised by Formal Logic const.i.tute a very meagre selection from the list of propositions actually used in judgment and reasoning.

Those Logicians who explicitly quantify the predicate obtain, in all, eight forms of proposition according to Quant.i.ty and Quality:

U. Toto-total Affirmative -- All X is all Y.

A. Toto-partial Affirmative -- All X is some Y.

Y. Parti-total Affirmative -- Some X is all Y.

I. Parti-partial Affirmative -- Some X is some Y.

E. Toto-total Negative -- No X is any Y.

?. Toto-partial Negative -- No X is some Y.

O. Parti-total Negative -- Some X is not any Y.

?. Parti-partial Negative -- Some X is not some Y.

Here A. I. E. O. correspond with those similarly symbolised in the usual list, merely designating in the predicates the quant.i.ty which was formerly treated as implicit.

-- 4. As to Relation, propositions are either Categorical or Conditional.

A Categorical Proposition is one in which the predicate is directly affirmed or denied of the subject without any limitation of time, place, or circ.u.mstance, extraneous to the subject, as _All men in England are secure of justice_; in which proposition, though there is a limitation of place ('in England'), it is included in the subject. Of this kind are nearly all the examples that have yet been given, according to the form _S is P_.

A Conditional Proposition is so called because the predication is made under some limitation or condition not included in the subject, as _If a man live in England, he is secure of justice_. Here the limitation 'living in England' is put into a conditional sentence extraneous to the subject, 'he,' representing any man.

Conditional propositions, again, are of two kinds--Hypothetical and Disjunctive. Hypothetical propositions are those that are limited by an explicit conditional sentence, as above, or thus: _If Joe Smith was a prophet, his followers have been unjustly persecuted_. Or in symbols thus:

If A is, B is; If A is B, A is C; If A is B, C is D.

Disjunctive propositions are those in which the condition under which predication is made is not explicit but only implied under the disguise of an alternative proposition, as _Joe Smith was either a prophet or an impostor_. Here there is no direct predication concerning Joe Smith, but only a predication of one of the alternatives conditionally on the other being denied, as, _If Joe Smith was not a prophet he was an impostor_; or, _If he was not an impostor, he was a prophet_. Symbolically, Disjunctives may be represented thus:

A is either B or C, Either A is B or C is D.

Formally, every Conditional may be expressed as a Categorical. For our last example shows how a Disjunctive may be reduced to two Hypotheticals (of which one is redundant, being the contrapositive of the other; see chap. vii. -- 10). And a Hypothetical is reducible to a Categorical thus: _If the sky is clear, the night is cold_ may be read--_The case of the sky being clear is a case of the night being cold_; and this, though a clumsy plan, is sometimes convenient. It would be better to say _The sky being clear is a sign of the night being cold_, or a condition of it.

For, as Mill says, the essence of a Hypothetical is to state that one clause of it (the indicative) may be inferred from the other (the conditional). Similarly, we might write: _Proof of Joe Smith's not being a prophet is a proof of his being an impostor_.

This turning of Conditionals into Categoricals is called a Change of Relation; and the process may be reversed: _All the wise are virtuous_ may be written, _If any man is wise he is virtuous_; or, again, _Either a man is not-wise or he is virtuous_. But the categorical form is usually the simplest.

If, then, as subst.i.tutes for the corresponding conditionals, categoricals are formally adequate, though sometimes inelegant, it may be urged that Logic has nothing to do with elegance; or that, at any rate, the chief elegance of science is economy, and that therefore, for scientific purposes, whatever we may write further about conditionals must be an ugly excrescence. The scientific purpose of Logic is to a.s.sign the conditions of proof. Can we, then, in the conditional form prove anything that cannot be proved in the categorical? Or does a conditional require to be itself proved by any method not applicable to the Categorical? If not, why go on with the discussion of Conditionals?

For all laws of Nature, however stated, are essentially categorical. 'If a straight line falls on another straight line, the adjacent angles are together equal to two right angles'; 'If a body is unsupported, it falls'; 'If population increases, rents tend to rise': here 'if' means 'whenever' or 'all cases in which'; for to raise a doubt whether a straight line is ever conceived to fall upon another, whether bodies are ever unsupported, or population ever increases, is a superfluity of scepticism; and plainly the hypothetical form has nothing to do with the proof of such propositions, nor with inference from them.

Still, the disjunctive form is necessary in setting out the relation of contradictory terms, and in stating a Division (chap. xxi.), whether formal (_as A is B or not-B_) or material (as _Cats are white, or black, or tortoisesh.e.l.l, or tabby_). And in some cases the hypothetical form is useful. One of these occurs where it is important to draw attention to the condition, as something doubtful or especially requiring examination. _If there is a resisting medium in s.p.a.ce, the earth will fall into the sun; If the Corn Laws are to be re-enacted, we had better sell railways and buy land_: here the hypothetical form draws attention to the questions whether there is a resisting medium in s.p.a.ce, whether the Corn Laws are likely to be re-enacted; but as to methods of inference and proof, the hypothetical form has nothing to do with them.

The propositions predicate causation: _A resisting medium in s.p.a.ce is a condition of the earth's falling into the sun; A Corn Law is a condition of the rise of rents, and of the fall of railway profits_.

A second case in which the hypothetical is a specially appropriate form of statement occurs where a proposition relates to a particular matter and to future time, as _If there be a storm to-morrow, we shall miss our picnic_. Such cases are of very slight logical interest. It is as exercises in formal thinking that hypotheticals are of most value; inasmuch as many people find them more difficult than categoricals to manipulate.

In discussing Conditional Propositions, the conditional sentence of a Hypothetical, or the first alternative of a Disjunctive, is called the Antecedent; the indicative sentence of a Hypothetical, or the second alternative of a Disjunctive, is called the Consequent.

Hypotheticals, like Categoricals, have been cla.s.sed according to Quant.i.ty and Quality. Premising that the quant.i.ty of a Hypothetical depends on the quant.i.ty of its Antecedent (which determines its limitation), whilst its quality depends on the quality of its consequent (which makes the predication), we may exhibit four forms:

A. _If A is B, C is D;_ I. _Sometimes when A is B, C is D;_ E. _If A is B, C is not D;_ O. _Sometimes when A is B, C is not D._

But I. and O. are rarely used.

As for Disjunctives, it is easy to distinguish the two quant.i.ties thus:

A. _Either A is B, or C is D;_ I. _Sometimes either A is B or C is D._

But I. is rarely used. The distinction of quality, however, cannot be made: there are no true negative forms; for if we write--

_Neither is A B, nor C D,_